A hurricane warning for Fay, which earlier left five people dead in Haiti and the Dominican Republic, was issued from Havana eastward to Cuba's central Sancti Spiritus province.
Authorities in four Cuban provinces evacuated nearly 5,000 residents from low-lying communities and pulled fishing boats out of the water, while setting up temporary shelters and food distribution centers.
Officials suspended traditional carnival celebrations in the central province of Cienfuegos and state television showed images of workers moving equipment for precautionary reasons at ports and oil refineries. State media said authorities were ready to "protect" the 24,000 foreign tourists in the famous beach resort of Varadero, but provided no more details.
Winds damaged the roofs of some homes in little-populated areas and water accumulated on roads and highways, but there were no reports of major flooding.
Jose Rubiera, Cuba's chief meteorologist, said Fay had brought gusts of wind of up to 70 mph (110 kph) as its center roared close to two coastal communities on the island's southern tip.
The storm had slowed late Sunday and was expected to gain force. Rubiera told state television it could get close to reaching hurricane strength before crossing over Cuba.
In the city of Niquero, near the southern coast and one of the hardest-hit areas, authorities converted a hotel into a shelter for evacuees.
"It's raining intensely, but the wind comes and goes," said a receptionist at the Hotel Niquero, who said he was not authorized to have his name appear in the foreign press.
Officials also suspended some ferry service on Isla de la Juventud, an island off Cuba's southern coast. In the southeastern province of Granma, a banana plantation sustained minor flooding and storm winds damaged some homes, state media reported.
At 8 p.m. EDT (2100 GMT) on Sunday, Fay was centered about 200 miles (330 kilometers) southeast of Havana and 265 miles (430 kilometers) south-southeast of Key West, Florida, according to the U.S. National Hurricane Center in Miami.
It had maximum sustained winds near 50 mph (85 kph), was moving somewhat erratically west-northwest at 10 mph (16 kph). It was expected to turn to the northwest by late Monday, and was also expected to keep strengthening gradually.
Florida declared a state of emergency and a light stream of traffic headed out of Key West on Sunday as officials urged visitors to leave the string of low-lying islands.
Most of the islands sit at sea level and could be flooded by Fay's storm surge.
A tropical storm watch was in effect for much of southern Florida. A watch means storm conditions are possible within 36 hours.
The hurricane center said the storm could approach hurricane strength as it nears the Florida Keys late Monday or early Tuesday, possibly making landfall along the western coast of Florida.